Volume and surface area of 3D shapes
Volume is measured in cubic units and surface area in square units. S is the total surface area including any flat bases unless the entry says otherwise.
| Solid | Inputs | Formulas |
|---|---|---|
| Cube | Side | V = s³; S = 6s² |
| Rectangular prism | Length, Width, Height | V = lwh; S = 2(lw + lh + wh) |
| Sphere | Radius | V = 4πr³/3; S = 4πr² |
| Hemisphere | Radius | V = 2πr³/3; S = 3πr² (including base) |
| Cylinder | Radius, Height | V = πr²h; S = 2πr(r+h) |
| Cone | Radius, Perpendicular height | V = πr²h/3; S = πr(r+√(r²+h²)) |
| Conical frustum | Bottom radius, Top radius, Perpendicular height | V = πh(R²+Rr+r²)/3 |
| Square pyramid frustum | Bottom base side, Top base side, Perpendicular height | V = h(a² + ab + b²)/3; S = a² + b² + 2(a + b)·s, s = √(h² + ((a − b)/2)²) |
| Square pyramid | Base side, Perpendicular height | V = s²h/3 |
| Rectangular pyramid | Base length, Base width, Perpendicular height | V = lwh/3 |
| Ellipsoid | Semi-axis a, Semi-axis b, Semi-axis c | V = 4πabc/3 |
| Capsule | Radius, Straight cylinder length | V = πr²(h+4r/3); S = 2πr(2r+h) |
| Hollow cylinder | Outer radius, Inner radius, Height | V = π(R²−r²)h |
| Triangular prism — three triangle sides | Triangle side a, Triangle side b, Triangle side c, Prism length | End area T = √(s(s−a)(s−b)(s−c)); V = T·L; S = 2T + (a + b + c)L |
| Hollow cylinder — outer radius and wall thickness | Outer radius, Wall thickness, Height | r = R − t; V = π(R² − r²)h |
| Triangular prism | Triangle base, Triangle perpendicular height, Prism length | V = base × triangle height × length / 2 |
Sources: OpenStax: Contemporary Mathematics §10.7 Volume and Surface Area
Cavalieri's principle and prism rules
Any prism or cylinder has volume V = (base area) × (height), whatever the shape of its base. Any pyramid or cone with the same base and height has exactly one third of that volume: V = (base area) × h ÷ 3.
Scaling every length of a solid by a factor k multiplies its surface area by k² and its volume by k³.